A New Kind of High-Order Multi-step Schemes for Forward Backward Stochastic Differential Equations

In this work, we concern with the high order numerical methods for coupled forward-backward stochastic differential equations (FBSDEs). Based on the FBSDEs theory, we derive two reference ordinary differential equations (ODEs) from the backward SDE, which contain the conditional expectations and their derivatives. Then, our high order multi-step schemes are obtained by carefully approximating the derivatives and the conditional expectations in the reference ODEs. Motivated by the local property of the generator of diffusion processes, the Euler method is used to solve the forward SDE, however, it is noticed that the numerical solution of the backward SDE is still of high order accuracy. Such results are obviously promising: on one hand, the use of Euler method (for the forward SDE) can dramatically simplifies the entire computational scheme, and on the other hand, one might be only interested in the solution of the backward SDE in many real applications such as option pricing. Several numerical experiments are carried out to demonstrate the effectiveness of the numerical method

A generalized scheme for BSDEs based on derivative approximation and its error estimates

In this paper we propose a generalized numerical scheme for backward stochastic differential equations(BSDEs). The scheme is based on approximation of derivatives via Lagrange interpolation. By changing the distribution of sample points used for interpolation, one can get various numerical schemes with different stability and convergence order. We present a condition for the distribution of sample points to guarantee the convergence of the scheme.Comment: 11 pages, 1 table. arXiv admin note: text overlap with arXiv:1808.0156

High order numerical schemes for second-order FBSDEs with applications to stochastic optimal control

This is one of our series papers on multistep schemes for solving forward backward stochastic differential equations (FBSDEs) and related problems. Here we extend (with non-trivial updates) our multistep schemes in [W. Zhao, Y. Fu and T. Zhou, SIAM J. Sci. Comput., 36 (2014), pp. A1731-A1751.] to solve the second order FBSDEs (2FBSDEs). The key feature of the multistep schemes is that the Euler method is used to discrete the forward SDE, which dramatically reduces the entire computational complexity. Moreover, it is shown that the usual quantities of interest (e.g., the solution tuple $(Y_t, Z_t, A_t, \Gamma_t)$ in the 2FBSDEs) are still of high order accuracy. Several numerical examples are given to show the effective of the proposed numerical schemes. Applications of our numerical schemes for stochastic optimal control problems are also presented

A Review of Tree-based Approaches to solve Forward-Backward Stochastic Differential Equations

In this work, we study solving (decoupled) forward-backward stochastic differential equations (FBSDEs) numerically using the regression trees. Based on the general theta-discretization for the time-integrands, we show how to efficiently use regression tree-based methods to solve the resulting conditional expectations. Several numerical experiments including high-dimensional problems are provided to demonstrate the accuracy and performance of the tree-based approach. For the applicability of FBSDEs in financial problems, we apply our tree-based approach to the Heston stochastic volatility model, the high-dimensional pricing problems of a Rainbow option and an European financial derivative with different interest rates for borrowing and lending

Adapted θ\theta-Scheme and Its Error Estimates for Backward Stochastic Differential Equations

In this paper we propose a new kind of high order numerical scheme for backward stochastic differential equations(BSDEs). Unlike the traditional $\theta$-scheme, we reduce truncation errors by taking $\theta$ carefully for every subinterval according to the characteristics of integrands. We give error estimates of this nonlinear scheme and verify the order of scheme through a typical numerical experiment.Comment: 18 pages, 3 tables, 1 figur

Multistep schemes for solving backward stochastic differential equations on GPU

The goal of this work is to parallelize the multistep scheme for the numerical approximation of the backward stochastic differential equations (BSDEs) in order to achieve both, a high accuracy and a reduction of the computation time as well. In the multistep scheme the computations at each grid point are independent and this fact motivates us to select massively parallel GPU computing using CUDA. In our investigations we identify performance bottlenecks and apply appropriate optimization techniques for reducing the computation time, using a uniform domain. Finally, some examples with financial applications are provided to demonstrate the achieved acceleration on GPUs.Comment: 24 pages, 4 figures, 10 table

Multilevel approximation of backward stochastic differential equations

We develop a multilevel approach to compute approximate solutions to backward differential equations (BSDEs). The fully implementable algorithm of our multilevel scheme constructs sequential martingale control variates along a sequence of refining time-grids to reduce statistical approximation errors in an adaptive and generic way. We provide an error analysis with explicit and non-asymptotic error estimates for the multilevel scheme under general conditions on the forward process and the BSDE data. It is shown that the multilevel approach can reduce the computational complexity to achieve precision $\epsilon$, ensured by error estimates, essentially by one order (in $\epsilon^{-1}$) in comparison to established methods, which is substantial. Computational examples support the validity of the theoretical analysis, demonstrating efficiency improvements in practice

Improved error bounds for quantization based numerical schemes for BSDE and nonlinear filtering

We take advantage of recent and new results on optimal quantization theory to improve the quadratic optimal quantization error bounds for backward stochastic differential equations (BSDE) and nonlinear filtering problems. For both problems, a first improvement relies on a Pythagoras like Theorem for quantized conditional expectation. While allowing for some locally Lipschitz functions conditional densities in nonlinear filtering, the analysis of the error brings into playing a new robustness result about optimal quantizers, the so-called distortion mismatch property: $L^r$-quadratic optimal quantizers of size $N$ behave in $L^s$ in term of mean error at the same rate $N^{-\frac 1d}$, $0<s< r+d$

Linear multi-step schemes for BSDEs

We study the convergence rate of a class of linear multi-step methods for BSDEs. We show that, under a sufficient condition on the coefficients, the schemes enjoy a fundamental stability property. Coupling this result to an analysis of the truncation error allows us to design approximation with arbitrary order of convergence. Contrary to the analysis performed in \cite{zhazha10}, we consider general diffusion model and BSDEs with driver depending on $z$. The class of methods we consider contains well known methods from the ODE framework as Nystrom, Milne or Adams methods. We also study a class of Predictor-Correctot methods based on Adams methods. Finally, we provide a numerical illustration of the convergence of some methods.Comment: 30 pages, 2 figure

Efficient spectral sparse grid approximations for solving multi-dimensional forward backward SDEs

This is the second part in a series of papers on multi-step schemes for solving coupled forward backward stochastic differential equations (FBSDEs). We extend the basic idea in our former paper [W. Zhao, Y. Fu and T. Zhou, SIAM J. Sci. Comput., 36 (2014), pp. A1731-A1751] to solve high-dimensional FBSDEs, by using the spectral sparse grid approximations. The main issue for solving high dimensional FBSDEs is to build an efficient spatial discretization, and deal with the related high dimensional conditional expectations and interpolations. In this work, we propose the sparse grid spatial discretization. We use the sparse grid Gaussian-Hermite quadrature rule to approximate the conditional expectations. And for the associated high dimensional interpolations, we adopt an spectral expansion of functions in polynomial spaces with respect to the spatial variables, and use the sparse grid approximations to recover the expansion coefficients. The FFT algorithm is used to speed up the recovery procedure, and the entire algorithm admits efficient and high accurate approximations in high-dimensions, provided that the solutions are sufficiently smooth. Several numerical examples are presented to demonstrate the efficiency of the proposed methods